Capacitated Vehicle Routing with Nonuniform Speeds

Abstract

The capacitated vehicle routing problem (CVRP) involves distributing identical items from a depot to a set of demand locations using a single capacitated vehicle. We introduce the heterogeneous capacitated vehicle routing problem, a generalization of CVRP to the setting of multiple vehicles having nonuniform speeds, and present for it a constant-factor approximation algorithm. Our main contribution is an approximation algorithm for the heterogeneous traveling salesman problem, which is the special case of heterogeneous CVRP with uncapacitated vehicles. Given a metric denoting distances between vertices, a depot r containing k vehicles having respective speeds [Formula: see text], the objective in heterogeneous TSP is to find a tour for each vehicle (starting and ending at r) so that every vertex is covered in some tour and the maximum completion time is minimized; the completion time of a vehicle is the distance traveled divided by its speed. Our algorithm relies on a new approximate minimum spanning tree construction called Level-Prim, which is related to but different from Light Approximate Shortest-path Trees. We also extend the widely used tour-splitting technique to nonuniform speeds, using ideas from the 2-approximation algorithm for scheduling in unrelated machines.